Also, solving problems from the above books or from this set , for example is recommended. Kallenberg's inimitable book Foundations of probability contains material worth other books, but my feeling is that it is best for the second reading rather than the first. Basic, but slightly more advanced Some of the topics that are basic but not adequately covered in the Durrett's book can be read from the following list.
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This is a topic-wise listing, rather than a recommendation to read all the books cover to cover! Weak convergence theory from Billingsley's Weak convergence of probability measures or K. Parthasarathy's Probability measures on metric spaces. Stochastic Calculus Karatzas and Shreve Brownian motion and stochastic calculus Large deviations, at least Sanov's theorem Dembo and Zeitouni's Large deviations: techniques and applications and concentration of measure first three chapters of Boucheron, Lugosi and Massart's Concentration inequalities: a non-asymptotic theory of independence.
Discrete probability: The amazingly beautiful book Probability on trees and networks by Lyons and Peres, Grimmett's Probability on graphs and the superb lecture notes by Sebastain Roch and the well-known book The probabilistic method by Alon and Spencer. More about Brownian motion, its connections to PDE etc. Varadhan's TIFR lecture notes Diffusions and partial differential equations Markov processes Liggett's Continuous time Markov processes: an introduction , just the first chapters would be good Mixing times of Markov chains book of the same name by Levin, Peres and Wilmer, a few chapters give sufficiently many of the ideas Gaussian Hilbert spaces Janson's Gaussian Hilbert spaces , initial part, is very good for learning Wiener chaos decomposition, Wick formula etc.
It is also good to study stationary Gaussian processes on the line, and some basics of continuity of Gaussian processes Basics of information theory Cover and Thomas have written a good book. Basic notions of entropy, relative entropy etc are essential for probabilists Point processes, random measures. Two upcoming books: Kallenberg's Random measures, theory and applications which may be an update of his old book titled Random measures and Lectures on the Poisson Process by Last and Penrose.
And here is a great presentation of Oded Schramm about his work with Garban and Peter, earlier work with Steiff and some basic things about noise sensitivity. The Percolation Fourier Spectrum Here, we establish universal Gaussian fluctuations for traces of general possibly discrete, as Bernoulli matrix ensembles.
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- Proceedings Volume 15 Contents?
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- SMC Colloquium | KTH.
- Lectures on noise sensitivity and percolation!
The distribution of S under this measure encodes important aspects of the behaviour of f under noise and under the natural dynamics on the discrete cube. We derive consequences for the behavior of percolation under noise and for exceptional times of dynamical percolation. Background in percolation theory will not be assumed. Share this: Facebook Reddit Twitter. Like this: Like Loading Bookmark the permalink.
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Subgaussian concentration and rates of convergence in directed polymers. First passage percolation has sublinear distance variance. MR Bobkov, S. Concentration inequalities using the entropy method.
Concentration inequalities. A nonasymptotic theory of independence. With a foreword by Michel Ledoux. Oxford University Press, Oxford, MR N. Chatterjee, Sourav.
Related Lectures on Noise Sensitivity and Percolation
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